Sweedler's Hopf algebra

In mathematics, introduced an example of an infinite-dimensional Hopf algebra, and Sweedler's Hopf algebra H4 is a certain 4-dimensional quotient of it  that is neither commutative nor cocommutative.

Definition
The following infinite dimensional Hopf algebra was introduced by. The Hopf algebra is generated as an algebra by three elements x, g and g-1.

The coproduct Δ is given by
 * Δ(g) = g ⊗g, Δ(x) = 1⊗x + x ⊗g

The antipode S is given by
 * S(x) = –x g−1, S(g) = g−1

The counit ε is given by
 * ε(x)=0, ε(g) = 1

Sweedler's 4-dimensional Hopf algebra H4 is the quotient of this by the relations
 * x2 = 0,  g2 = 1,  gx = –xg

so it has a basis 1, x, g, xg. Note that Montgomery describes a slight variant of this Hopf algebra using the opposite coproduct, i.e. the coproduct described above composed with the tensor flip on H4⊗H4. This Hopf algebra is isomorphic to the Hopf algebra described here by the Hopf algebra homomorphism $$g\mapsto g$$ and $$x\mapsto gx$$.

Sweedler's 4-dimensional Hopf algebra is a quotient of the Pareigis Hopf algebra, which is in turn a quotient of the infinite dimensional Hopf algebra.